数学物理学报(英文版) ›› 2016, Vol. 36 ›› Issue (2): 325-333.doi: 10.1016/S0252-9602(16)30002-9

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DIFFERENTIAL OPERATORS OF INFINITE ORDER IN THE SPACE OF RAPIDLY DECREASING SEQUENCES

M. MALDONADO, J. PRADA, M. J. SENOSIAIN   

  1. Department of Mathematics, Salamanca University, Salamanca, Spain
  • 收稿日期:2014-02-18 修回日期:2015-05-07 出版日期:2016-04-25 发布日期:2016-04-25
  • 通讯作者: J. PRADA,E-mail:prada@usal.es E-mail:prada@usal.es
  • 作者简介:M. MALDONADO,E-mail:cordero@usal.es;M.J. SENOSIAIN,E-mail:idiazabal@usal.es
  • 基金资助:

    Supported by MIMECO under project MTM2012-38445.

DIFFERENTIAL OPERATORS OF INFINITE ORDER IN THE SPACE OF RAPIDLY DECREASING SEQUENCES

M. MALDONADO, J. PRADA, M. J. SENOSIAIN   

  1. Department of Mathematics, Salamanca University, Salamanca, Spain
  • Received:2014-02-18 Revised:2015-05-07 Online:2016-04-25 Published:2016-04-25
  • Contact: J. PRADA,E-mail:prada@usal.es E-mail:prada@usal.es
  • Supported by:

    Supported by MIMECO under project MTM2012-38445.

摘要:

We consider the space of rapidly decreasing sequences s and the derivative operator D defined on it. The object of this article is to study the equivalence of a differential operator of infinite order; that is φ(D)=φkDk.φk constant numbers an a power of D. Dn, meaning, is there a isomorphism X (from s onto s) such that Xφ(D)=DnX?. We prove that if φ(D) is equivalent to Dn, then φ(D) is of finite order, in fact a polynomial of degree n. The question of the equivalence of two differential operators of finite order in the space s is addressed too and solved completely when n=1.

关键词: Ordinary differential operators, sequence spaces, operators on function spaces

Abstract:

We consider the space of rapidly decreasing sequences s and the derivative operator D defined on it. The object of this article is to study the equivalence of a differential operator of infinite order; that is φ(D)=φkDk.φk constant numbers an a power of D. Dn, meaning, is there a isomorphism X (from s onto s) such that Xφ(D)=DnX?. We prove that if φ(D) is equivalent to Dn, then φ(D) is of finite order, in fact a polynomial of degree n. The question of the equivalence of two differential operators of finite order in the space s is addressed too and solved completely when n=1.

Key words: Ordinary differential operators, sequence spaces, operators on function spaces

中图分类号: 

  • 47E05